THE ASYMPTOTIC DISTRIBUTION OF CIRCLES IN THE ORBITS OF KLEINIAN GROUPS

Transcription

1 THE ASYMPTOTIC DISTRIBUTION OF CIRCLES IN THE ORBITS OF KLEINIAN GROUPS HEE OH AND NIMISH SHAH Abstract. Let P be a locally finite circle packing in the plane C invariant under a non-elementary Kleinian group and with finitely many -orbits. When is geometrically finite, we construct an explicit Borel measure on C which describes the asymptotic distribution of small circles in P, assuming that either the critical exponent of is strictly bigger than 1 or P does not contain an infinite bouquet of tangent circles glued at a parabolic fixed point of. Our construction also works for P invariant under a geometrically infinite group, provided admits a finite Bowen-Margulis-Sullivan measure and the -skinning size of P is finite. Some concrete circle packings to which our result applies include Apollonian circle packings, Sierpinski curves, Schottky dances, etc. 1. Introduction A circle packing in the plane C is simply a union of circles (here a line is regarded as a circle of infinite radius). As we allow circles to intersect with each other, our definition of a circle packing is more general than the conventional definition of a circle packing. For a given circle packing P in the plane, we are interested in counting and distribution of small circles in P. A natural size of a circle is measured by its radius. We will use the curvature of a circle, that is, the reciprocal of its radius, instead. We suppose that P is locally finite in the sense that for any T > 1, there are only finitely many circles in P of curvature at most T in any fixed bounded subset of C. Geometrically, P is locally finite if there is no infinite sequence of circles in P converging to a fixed circle. For instance, if the circles of P have disjoint interiors as in Fig. 1, P is locally finite. For a bounded subset E of C and T > 1, we set N T (P, E) := #{C P : C E, Curv(C) < T } where Curv(C) denotes the curvature of a circle C. The local finiteness assumption on P implies that N T (P, E) <. Our question is then if there exists a Borel measure ω P on C such that for all nice Borel subsets Oh is partially supported by NSF grant Shah is partially supported by NSF grant

2 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 2 Figure 1. Apollonian circle packing and Sierpinski curve (by C. McMullen) E1, E2 C, NT (P, E1 ) ωp (E1 ) T, NT (P, E2 ) ωp (E2 ) assuming NT (P, E2 ) > 0 and ωp (E2 ) > 0. Our main theorem applies to a very general packing P, provided P is invariant under a non-elementary (i.e., non virtually-abelian) Kleinian group satisfying certain finiteness conditions. Recall that a Kleinian group is a discrete subgroup of G := PSL2 (C) and G acts on the extended complex plane C = C { } by Mo bius transformations: az + b a b z= c d cz + d where a, b, c, d C with ad bc = 1 and z C. A Mo bius transformation maps a circle to a circle and by the Poincare extension, G can be identified with the group of all orientation preserving isometries of H3. Considering the upper-half space model H3 = {(z, r) : z C, r > 0}, the geometric boundary (H3 ) is naturally identified with C. For a Kleinian group, we denote by Λ() C its limit set, that is, the set of accumulation points of an orbit of in C, and by 0 δ 2 its critical exponent. For non-elementary, it is known that δ > 0. Let {νx : x H3 } be a -invariant conformal density of dimension δ on Λ(), which exists by the work of Patterson [23] and Sullivan [30]. In order to present our main theorem on the asymptotic of NT (P, E) we introduce two invariants associated to and P. The first one is a Borel measure on C depending only on.

3 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 3 Definition 1.1. Define a Borel measure ω on C: for ψ C c (C) ω (ψ) = ψ(z)e δ β z(x,(z,1)) dν x (z) z C where x H 3 and β z (x 1, x 2 ) is the signed distance between the horospheres based at z C and passing through x 1, x 2 H 3. By the conformal property of {ν x }, ω is well-defined independent of the choice of x H n. We have a simple formula: for j = (0, 1) H 3, dω = ( z 2 + 1) δ dν j. For a vector u in the unit tangent bundle T 1 (H 3 ), denote by u + Ĉ (resp. u Ĉ) the forward (resp. backward) end point of the geodesic determined by u. On the contracting horosphere H (j) T 1 (H 3 ) consisting of upward unit normal vectors on the horizontal plane {(z, 1) : z C}, the normal vector based at (z, 1) is mapped to z via the map u u. Under this correspondence, the measure ω on C is equal to the density of the Burger- Roblin measure m BR (see Def. 2.3) on H (j). The second invariant is a number in [0, ] measuring a certain size of P. Definition 1.2 (The -skinning size of P). For a circle packing P invariant under, define 0 sk (P) as follows: sk (P) := e δ β s + (x,π(s)) dν x (s + ) i I s Stab (C i )\C i where x H 3, π : T 1 (H 3 ) H 3 is the canonical projection, {C i : i I} is a set of representatives of -orbits in P, C i T 1 (H 3 ) is the set of unit normal vectors to the convex hull Ĉ i of C i and Stab (C i ) denotes the setwise stabilizer of C i in. Again by the conformal property of {ν x }, the definition of sk (P) is independent of the choice of x and the choice of representatives {C i }. We remark that the value of sk (P) can be zero or infinite in general and we do not assume any condition on Stab (C i ) s (they may be trivial). We denote by m BMS the Bowen-Margulis-Sullivan measure on the unit tangent bundle T 1 (\H 3 ) associated to the density {ν x } (Def. 2.2). When is geometrically finite, i.e., admits a finite sided fundamental domain in H 3, Sullivan showed that m BMS < [31] and that δ is equal to the Hausdorff dimension of the limit set Λ() [30]. A point in Λ() is called a parabolic fixed point of if it is fixed by a parabolic element of. Definition 1.3. By an infinite bouquet of tangent circles glued at a point ξ C, we mean a union of two collections, each consisting of infinitely many pairwise internally tangent circles with the common tangent point ξ and their radii tending to 0, such that the circles in each collection are externally tangent to the circles in the other at ξ (see Fig. 2).

4 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 4 Figure 2. Infinite bouquet of tangent circles Theorem 1.4. Let P be a locally finite circle packing in C invariant under a non-elementary geometrically finite group and with finitely many -orbits. If δ 1, we further assume that P does not contain an infinite bouquet of tangent circles glued at a parabolic fixed point of. Then sk (P) < and for any bounded Borel subset E of C with ω ( (E)) = 0, N T (P, E) lim T T δ = sk (P) δ m BMS ω (E). If P has infinitely many circles, then sk (P) > 0. Remark 1.5. (1) Given a finite collection {C 1,, C m } of circles in the plane C and a non-elementary geometrically finite group < PSL 2 (C), Theorem 1.4 applies to P := m i=1 (C i), provided P contains neither infinitely many circles converging to a fixed circle nor any infinite bouquet of tangent circles. (2) In the case when δ 1 and P contains an infinite bouquet of tangent circles glued at a parabolic fixed point of, we have sk (P) = [19]. In that case if the interior of E intersects Λ() non-trivially, the growth order of N T (P, E) is T log T if δ = 1, and it is T if δ < 1 [21]. (3) We note that the asymptotic of N T (P, E) depends only on, except for the -skinning size of P. This is rather surprising in view of the fact that there are circle packings with completely different configurations but invariant under the same group. (4) Theorem 1.4 implies that the asymptotic distribution of small circles in P is completely determined by the measure ω : for any bounded Borel sets E 1, E 2 with ω (E 2 ) > 0 and ω ( (E i )) = 0, i = 1, 2, as T, N T (P, E 1 ) N T (P, E 2 ) ω (E 1 ) ω (E 2 ). (5) Suppose that all circles in P can be oriented so that they have disjoint interiors whose union is equal to the domain of discontinuity Ω() := Ĉ Λ(). If either P is bounded or is a parabolic fixed point for, then δ is equal to the circle packing exponent e P defined

5 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 5 as: e P = inf{s : C P Curv s < } = sup{s : C P Curv(C) s = }. This was proved by Parker [22] extending the earlier works of Boyd [3] and Sullivan [31] on bounded Apollonian circle packings. In the proof of Theorem 1.4, the geometric finiteness assumption on is used only to ensure the finiteness of the Bowen-Margulis-Sullivan measure. We have the following more general theorem: m BMS Theorem 1.6. Let P be a locally finite circle packing invariant under a non-elementary Kleinian group and with finitely many -orbits. Suppose that m BMS < and sk (P) <. Then for any bounded Borel subset E of C with ω ( (E)) = 0, N T (P, E) lim T T δ If P is infinite, then sk (P) > 0. = sk (P) δ m BMS ω (E). Since there is a large class of geometrically infinite groups with m BMS < [24], Theorem 1.6 is not subsumed by Theorem 1.4. We remark that the condition on the finiteness of m BMS implies that the density {ν x } is determined uniquely up to homothety (see [26, Coro. 1.8]). Remark 1.7. (1) The assumption of m BMS < implies that ν x (and hence ω ) is atom-free [26, Sec. 1.5], and hence the above theorem works for any bounded Borel subset E intersecting Λ() only at finitely many points. (2) It is not hard to show that is Zariski dense in PSL 2 (C) considered as a real algebraic group if and only if Λ() is not contained in a circle in Ĉ. In such a case, any proper real subvariety of Ĉ has zero ν x -measure. This is shown in [7, Cor.1.4] for geometrically finite but its proof works equally well if ν x is -ergodic, which is the case when m BMS <. Hence Theorem 1.6 holds for any Borel subset E whose boundary is contained in a countable union of real algebraic curves. We now describe some concrete applications of Theorem Apollonian gasket. Three mutually tangent circles in the plane determine a curvilinear triangle, say, T. By a theorem of Apollonius of Perga (c. 200 BC), one can inscribe a unique circle into the triangle T, tangent to all of the three circles. This produces three more curvilinear triangles inside T and we inscribe a unique circle into each triangle. By continuing to add circles in this way, we obtain an infinite circle packing of T, called the Apollonian gasket for T, say, A (see Fig. 3).

6 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 6 Figure 3. Apollonian gasket Figure 4. Dual circles By adding all the circles tangent to three of the given ones, not only those within T, one obtains an Apollonian circle packing P := P(T ), which may be bounded or unbounded (cf. [10] [9], [27], [28], [12]). Fixing four mutually tangent circles in P, consider the four dual circles determined by the six intersection points (see Fig. 4 where the dotted circles are dual circles to the solid ones), and denote by P the intersection of PSL 2 (C) and the group generated by the inversions with respect to those dual circles. Then P is a geometrically finite Zariski dense subgroup of the real algebraic group PSL 2 (C) preserving P, and its limit set in Ĉ coincides with the residual set of P (cf. [12]). We denote by α the Hausdorff dimension of the residual set of P, which is known to be (8) according to McMullen [16]. Corollary 1.8. Let T be a curvilinear triangle determined by three mutually tangent circles and A the Apollonian gasket for T. Then for any Borel subset E T whose boundary is contained in a countable union of real algebraic curves, N T (E) lim T T α = sk P (P) α m BMS ω P (E) where N T (E) := #{C A : C E, Curv(C) < T } and P = P(T ). Either when P is bounded and E is the disk enclosed by the largest circle of P, or when P lies between two parallel lines and E is the whole period, P

7 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 7 Figure 5. Regions whose boundary intersects Λ() at finitely many points (background pictures are reproduced with permission from Indra s Pearls, by D.Mumford, C. Series and D. Wright, copyright Cambridge University Press 2002) it was proved in [12] that N T (P, E) c T α for some c > 0. This implies that 1 N T (T ) T α. The approach in [12] was based on the Descartes circle theorem in parameterizing quadruples of circles of curvature at most T as vectors of maximum norm at most T in the cone defined by the Descartes quadratic equation. We remark that the fact that α is strictly bigger than 1 was crucial in the proof of [12] as based on the L 2 -spectral theory of P \H Counting circles in the limit set Λ(). If X is a finite volume hyperbolic 3-manifold with totally geodesic boundary, then its fundamental group := π 1 (X) is geometrically finite and X is homeomorphic to \H 3 Ω() where Ω() := Ĉ Λ() is the domain of discontinuity [11]. The set Ω() is a union of countably many disjoint open disks in this case and has finitely many -orbits by the Ahlfors finiteness theorem [1]. Hence Theorem 1.4 applies to counting these open disks in Ω() with respect to the curvature. 1 means that the ratio of the two sides is between two uniform constants

8 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 8 Figure 6. Schottky dance (reproduced with permission from Indra s Pearls, by D.Mumford, C. Series and D. Wright, copyright Cambridge University Press 2002) For example, for the group generated by reflections in the sides of a unique regular tetrahedron whose convex core is bounded by four π 4 triangles and four right hexagons, Ω() is illustrated in the second picture in Fig. 1 (see [15, P.9] for details). This circle packing is called a Sierpinski curve, being homeomorphic to the well-known Sierpinski carpet [5]. Two pictures in Fig. 5 can be found in the beautiful book Indra s pearls by Mumford, Series and Wright (see P. 269 and P. 297 of [17]) where one can find many more circle packings to which our theorem applies. The book presents explicit geometrically finite Schottky groups whose limit sets are illustrated in Fig. 5. The boundaries of the shaded regions meet Λ() only at finitely many points. Hence our theorem applies to counting circles in these shaded regions Schottky dance. Another class of examples is obtained by considering the images of Schottky disks under Schottky groups. Take k 1 pairs of mutually disjoint closed disks {(D i, D i ) : 1 i k} in C and for each 1 i k, choose a Möbius transformation γ i which maps the interior of D i to the exterior of D i and the interior of D i to the exterior of D i. The group, say,, generated by {γ i : 1 i k} is called a Schottky group of genus k (cf. [13, Sec. 2.7]). The -orbit of the disks D i and D i s nests down onto the limit set Λ() which is totally disconnected. If we denote by P the

9 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 9 union 1 i k (C i ) (C i ) where C i and C i are the boundaries of D i and D i respectively, then P is locally finite, as the nesting disks become smaller and smaller. The common exterior of hemispheres above the initial disks D i and D i is a fundamental domain for in the upper half-space H3, and hence is geometrically finite. Since P contains no infinite bouquet of tangent circles, Theorem 1.4 applies to P; for instance, we can count circles in the picture in Fig. 6 ([17, Fig. 4.11]). On the structure of the proof. In [12], the counting problem for a bounded Apollonian circle packing was related to the equidistribution of expanding closed horospheres on the hyperbolic 3-manifold \H 3. For a general circle packing, there is no analogue of the Descartes circle theorem which made such a relation possible. The main idea in our paper is to relate the counting problem for a general circle packing P invariant under with the equidistribution of orthogonal translates of a closed totally geodesic surface in T 1 (\H 3 ). Let C 0 denote the unit circle centered at the origin and H the stabilizer of C 0 in PSL 2 (C). Thus H\G may be considered as the space of totally geodesic planes of H 3. The important starting point is to describe certain subset B T (E) in H\G so that the number of circles in the packing P := (C 0 ) of curvature at most T intersecting E can be interpreted as the number of points in B T (E) of a discrete -orbit on H\G. We then describe the weighted limiting distribution of orthogonal translates of an H-period (H )\H (which corresponds to a properly immersed hyperbolic surface which may be of infinite area) along these sets B T (E) in terms of the Burger-Roblin measure (Theorem 4.3) using the main result in [19] (see Thm. 2.5). To translate the weighted limiting distribution result into the asymptotic for N T (P, E), we relate the density of the Burger-Roblin measure of the contracting horosphere H (j) with the measure ω. A version of Theorem 1.4 in a weaker form, and some of its applications stated above were announced in [18]. We remark that the methods of this paper can be easily generalized to prove a similar result for a sphere packing in the n-dimensional Euclidean space invariant under a non-elementary discrete subgroup of Isom(H n+1 ). Acknowledgment. We would like to thank Curt McMullen for inspiring discussions. The applicability of our other paper [19] in the question addressed in this paper came up in the conversation of the first named author with him during her one month visit to Harvard in October, She thanks the Harvard mathematics department for the hospitality. We would also like to thank Yves Benoist for helpful discussions. 2. Expansion of a hyperbolic surface by orthogonal geodesic flow We use the following coordinates for the upper half space model for H 3 : H 3 = {z + rj = (z, r) : z C, r > 0}

10 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 10 where j = (0, 1). The isometric action of G = PSL 2 (C), via the Poincare extension of the linear fractional transformations, is explicitly given as the following (cf. [6]): (2.1) ( ) a b (z + rj) = (az + b)( c z + d) + a cr 2 c d cz + d 2 + c 2 r 2 + r cz + d 2 + c 2 r 2 j. In particular, the stabilizer of j is the following maximal compact subgroup of G: ( ) a b K := PSU(2) = { : a b ā 2 + b 2 = 1}. and We set A := {a t := N := {n z := ( ) ( ) e t/2 0 e iθ 0 0 e t/2 : t R}, M := { 0 e iθ : θ R} ( ) 1 z : z C}, N 0 1 = {n z := ( ) 1 0 : z C}. z 1 We can identify H 3 with G/K via the map g(j) gk. Denoting by X 0 T 1 (H 3 ) the upward unit normal vector based at j, we can also identify the unit tangent bundle T 1 (H 3 ) with G.X 0 = G/M: here g.x 0 is given by dλ(g)(x 0 ) where λ(g) : G/K G/K is the left translation λ(g)(g K) = gg K and dλ(g) is its derivative at j. The geodesic flow {g t } on T 1 (H 3 ) corresponds to the right translation by a t on G/M: g t (gm) = ga t M. For a circle C in C, denote by Ĉ its convex hull, which is the northern hemisphere above C. Set C 0 to be the unit circle in C centered at the origin. The set-wise stabilizer of Ĉ0 in G is given by ( ) 0 1 H = PSU(1, 1) PSU(1, 1) 1 0 where ( ) a b PSU(1, 1) = { : a b 2 b 2 = 1}. ā Note that H is equal to the stabilizer of C 0 in G and hence Ĉ0 can be identified with H/H K. We have the following generalized Cartan decomposition (cf. [29]): for A + = {a t : t 0}, G = HA + K in the sense that every element of g G can be written as g = hak, h H, a A +, k K and h 1 a 1 k 1 = h 2 a 2 k 2 implies that a 1 = a 2, h 1 = h 2 m and k 1 = m 1 k 2 for some m H K Z G (A) = M.

11 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 11 As X 0 is orthogonal to the tangent space T j (Ĉ0), H.X 0 = H/M corresponds to the set of unit normal vectors to Ĉ0, which we will denote by C 0. Note that C 0 has two connected components, depending on their directions. For t R, the set g t (C 0 ) = (H/M)a t = (Ha t M)/M corresponds to a union of two surfaces consisting of the orthogonal translates of Ĉ0 by distance t in each direction, both having the same boundary C 0. Let < G be a non-elementary discrete subgroup. As in the introduction, let {ν x : x H 3 } be a -invariant conformal density on Ĉ of dimension δ, that is, each ν x is a finite measure on Ĉ satisfying that for any x, y H3, z Ĉ and γ, γ ν x = ν γx ; and dν y dν x (z) = e δ β z(y,x). Here γ ν x (R) = ν x (γ 1 (R)) for a Borel subset R Ĉ and the Busemann function β z (y 1, y 2 ) is given by lim t d(y 1, ξ t ) d(y 2, ξ t ) for a geodesic ray ξ t toward z. For u T 1 (H 3 ), we define u + Ĉ (resp. u Ĉ) to be the forward (resp. backward) end point of the geodesic determined by u and π(u) H 3 to be the basepoint. Fixing o H 3, the map u (u +, u, t := β u (π(u), o)) is a homeomorphism between T 1 (H 3 ) and (Ĉ Ĉ {(ξ, ξ) : ξ Ĉ}) R. Definition 2.2. The Bowen-Margulis-Sullivan measure m BMS associated to {ν x } ([2], [14], [31]) is the measure on T 1 (\H 3 ) induced by the following -invariant measure on T 1 (H 3 ): for x H 3, d m BMS (u) = e δ β u + (x,π(u)) e δ β u (x,π(u)) dν x (u + )dν x (u )dt. By the conformal properties of {ν x }, this definition is independent of the choice of x H 3. We also denote by {m x : x H 3 } a G-invariant conformal density of dimension 2, which is unique up to homothety: each m x a finite measure on Ĉ which is invariant under Stab G(x) and dm x (z) = e 2βz(y,x) dm y (z) for any x, y H 3 and z Ĉ. Definition 2.3. The Burger-Roblin measure m BR associated to {ν x } and {m x } ([4], [26]) is the measure on T 1 (\H n ) induced by the following - invariant measure on T 1 (H n ): d m BR (u) = e 2β u + (x,π(u)) e δ β u (x,π(u)) dm x (u + )dν x (u )dt for x H 3. By the conformal properties of {ν x } and {m x }, this definition is independent of the choice of x H 3. For any circle C, let H C = {g G : gc = C} = {g G : gc = C }.

12 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 12 We consider the following two measures on C : Fix any x H 3, and let (2.4) dµ Leb C (s) := e 2β s + (x,π(s)) dm x (s) and dµ PS C (s) := e δ β s + (x,π(s)) dν x (s + ). These definitions are independent of the choice of x and µ Leb (resp. µ PS ) is C C left-invariant by H C (resp. H C )). Hence we may consider the measures µ Leb and µ PS on the quotient (H )\C. C C We denote by sk (C) the total mass of µ PS ; that is, C sk (C) := e δ β s + (x,π(s)) dν x (s + ). s H\C 0 In general, sk (C) may be zero or infinite. Theorem 2.5 ([19, Theorem 1.9]). Suppose that the natural projection map H C \Ĉ \H3 is proper. Assume that m BMS < and sk (C) <. Then for any ψ C c (\G/M), as t, e (2 δ )t ψ(sa t )dµ Leb s ( H C )\C C (s) sk (C) m BMS mbr (ψ). Moreover sk (C) > 0 if [ : H C ] =. Note that if m BMS <, then is of divergence type; that is, the Poincare series of diverges at δ. When is of divergence type, the - invariant conformal density {ν x } of dimension δ is unique up to homothety (see [26, Remark following Corollary 1.8]): explicitly ν x can be taken as the weak-limit as s δ + of the family of measures 1 ν x (s) := e sd(x,γj) δ γj. γ e sd(j,γj) Recall that g PSL 2 (C) is parabolic if and only if g has a unique fixed point in Ĉ. Theorem 2.6 ([19, Theorem 5.2]). Let be geometrically finite. Suppose that the natural projection map H C \Ĉ \H3 is proper. Then sk (C) < if and only if either δ > 1 or any parabolic fixed point of lying on C is fixed by a parabolic element of H C. Proof. Note that in the notation of [19, Theorem 5.2], if we put E = Ĉ, which is a complete totally geodesic submanifold of H 3 of codimension 1, then (π(ẽ)) = C, Ẽ = C, Ẽ = H C, and µ PS E = sk (C). Hence the conclusion is immediate. 3. Reformulation into the orbital counting problem on the space of hyperbolic planes Let G = PSL 2 (C) and < G be a non-elementary discrete subgroup. Let C be a circle in Ĉ and H C denote the set-wise stabilizer of C in G. It is clear: γ

13 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 13 Lemma 3.1. If (C) is infinite, then [ : H C ] =. Lemma 3.2. The following are equivalent: (1) A circle packing (C) is locally finite; (2) the natural projection map f : H C \Ĉ \H3 is proper; (3) H C \H C is discrete in H C \G. Proof. We observe that the properness of f is equivalent to the condition that only finitely many distinct hemispheres in (Ĉ) intersects a given compact subset of H 3. Note that any compact subset of H 3 is contained in a compact subset the form E [r 1, r 2 ] = {(z, r) : z E, r 1 r r 2 } for E C compact and 0 < r 1 < r 2 <, and that the radius of a circle in C is same as the height of its convex hull in H 3. Hence the properness of the map f is again equivalent to the condition that for any r > 0 and a compact subset E C, there are only finitely many distinct circles in (C) intersecting E and of radii at least r, that is, (C) being locally finite, proving the equivalence of (1) and (2). It is straightforward to verify that the properness of f and that of the projection map H C \C \ T 1 (H 3 ) are equivalent. Let X C C such that X + C = Ĉ. Let M C = {g G : gx C = X C }. Since Ĉ is the unique totally geodesic submanifold of H 3 orthogonal to X C, M C is contained in H C. We identify G/M C with T 1 (H 3 ) via gm C gx C. Since H/M C identifies with C, the canonical map H C \H C /M C \G/M C is proper. Since M C is compact, it follows that H C \H C \G is proper. Equivalently H C is closed in G (see [19] for the equivalence). As is countable, this is again equivalent to the discreteness of H C \H C in H C \G. This proves the equivalence of (2) and (3). Remark 3.3. If H C is a lattice in H C, then H C is closed in G ([25, 1]), and hence (C) is a locally finite circle packing. In this case, by [19, Theorem 1.11], we have sk (C) <. Proposition 3.4. Let ξ C be a parabolic fixed point of. Suppose that (C) does not contain an infinite bouquet of tangent circles glued at ξ. Then ξ is a parabolic fixed point for H C. Proof. Suppose that there exists a parabolic element γ H C fixing ξ C. By sending ξ to Ĉ by an element of G, we may assume that ξ = and γ acts as a translation on C. Since γc C and C is a circle passing through, we have that {γ k C : k Z} is an infinite collection of parallel lines. By sending back to the original ξ, we see that {γ k C : k Z} is an infinite bouquet of tangent circles glued at ξ Deduction of Theorem 1.4 from Theorem 1.6. We only need to ensure that sk (P) <, or equivalently, sk (C) < for each C P. By the assumption in Theorem 1.4, if ξ C is any parabolic fixed point of, then by Proposition 3.4, ξ is a parabolic fixed point for H C. Therefore by Theorem 2.6, sk (C) <.

14 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS Relating counting on a single -orbit to a set B T (E) H\G. In the rest of this section, let C 0 denote the unit circle in C centered at the origin and let H := Stab(Ĉ0). We follow notations from Section 2. We assume that (C 0 ) is a locally finite circle packing of C. Let E be a bounded subset in C and set N T ((C 0 ), E) := #{C (C 0 ) : C E, Curv(C) < T }. For s > 0, we set A + s := {a t : 0 t s}; A s := {a t : 0 t s}. For a subset E C, we set N E := {n z : z E}. Definition 3.5 (Definition of B T (E)). For E C and T > 1, we define the subset B T (E) of H\G to be the image of the set KA + log T N E = {ka t n z G : k K, 0 t < log T, z E} under the canonical projection G H\G. For a bounded circle C in C, C denotes the open disk enclosed by C. We will not need this definition for a line since there can be only finitely many lines intersecting a fixed bounded subset in a locally finite circle packing. Definition 3.6. For a given circle packing P, a bounded subset E C is said to be P-admissible if, for any bounded circle C P, C E implies C E, possibly except for finitely many circles. The following translation of N T ((C 0 ), E) as the number of points in [e] B T (E), where [e] = H H\G, is crucial in our approach: Proposition 3.7. If E is (C 0 )-admissible, there exists m 0 N such that for all T 1, #[e] B T (E) m 0 N T ((C 0 ), E) #[e] B T (E) + m 0. Proof. Observe that #[e] B T (E) = #{[γ] H\ : Hγ KA + log T N E } = #{[γ] / H : γhk N E A log T K } = #{γ(ĉ0) : γhk N E A log T K } where the second equality is obtained by taking the inverse. Since N E A log T j = {(z, r) H3 : T 1 < r 1, z E} and K is the stabilizer of j in G, it follows that #[e] B T (E) = #{γ(ĉ0) : γ(ĉ0) contains (z, r) with z E, T 1 < r 1}. By the admissibility assumption on E, we observe that γ(ĉ0) contains (z, r) with z E and T 1 < r 1 if and only if the center of γ(c 0 ) lies in E and the radius of γ(c 0 ) is greater than T 1, possibly except for finitely many number (say, m 0 ) of circles.

15 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS Uniform distribution along the family B T (E) and the Burger-Roblin measure We keep the notations C 0, H, K, M, A +, X 0, G, {m x : x H 3 }, etc., from section 2. Denoting by dm the probability invariant measure on M, (4.1) dh = dµ Leb (s)dm C 0 is a Haar measure on H = C 0 M as µleb is H-invariant, and the following C 0 defines a Haar measure on G: for g = ha r k HA + K, (4.2) dg = 4 sinh r cosh r dhdrdm j (k) where dm j (k) := dm j (k.x 0 + ). We denote by dλ the unique measure on H\G which is compatible with the choice of dg and dh: for ψ C c (G), ψ dg = ψ(h[g]) dhdλ[g]. G [g] H\G For a bounded set E C, recall that the set B T (E) in H\G is the image of the set h H KA + log T N E = {ka t n z G : k K, 0 t < log T, z E} under the canonical projection G H\G. The goal of this section is to deduce the following theorem 4.3 from Theorem 2.5: Theorem 4.3. Let be a non-elementary discrete subgroup of G. Suppose that m BMS < and sk (C 0 ) <. Suppose that the natural projection map H\Ĉ0 \H 3 is proper. Then for any bounded Borel subset E C and for any ψ C c (\G), we have 1 lim ψ(hg)dhdλ(g) = sk (C 0 ) T δ m BMS (ψ n ) dn T δ g B T (E) h H\H m BR n N E where ψ n C c (\G) M is given by ψ n (g) = m M ψ(gmn)dm and dn is the Lebesgue measure on N. In order to prove this result using Theorem 2.5, it is crucial to understand the shape of the set B T (E) in the HA + K decomposition of G. This is one of the important technical steps in the proof. On the shape of B T (E): Fix a left-invariant metric on G. For ɛ > 0, let U ɛ be the ɛ-ball around e in G. For a subset W of G, we set W ɛ = W U ɛ. Proposition 4.4. (1) If a t HKa s K for s > 0, then t s. (2) Given any ɛ > 0, there exists T 0 = T 0 (ɛ) such that {k K : a t k HKA + for some t > T 0 } K ɛ M.

16 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 16 Proof. Suppose a t = hk 1 a s k 2 for h H, k 1, k 2 K. We note that, as Aj is orthogonal to Ĉ0 and j Ĉ0, t = d(ĉ0, a t j) = d(ĉ0, hk 1 a s j) = d(ĉ0, k 1 a s j) d(j, k 1 a s j) = d(j, a s j) = s, proving the first claim. For the second claim, suppose a t k HKa s for some s 0. Then ka s a t HK. Applying both sides to j H 3, k(e s j) a t Ĉ 0. Now a t Ĉ 0 = e t Ĉ 0 is the northern hemisphere of Euclidean radius e t about 0 in H 3. On the other hand A j = (0, 1]j for A = {a s : s 0} and K ɛ {(0, 1]j} consists of geodesic rays in H 3 joining j and points of K ɛ (0) C. Now K ɛ (0) contains a disk of radius, say r ɛ > 0, centered at 0 in C, and hence K ɛ {(0, 1]j} contains a Euclidean half ball of radius r ɛ > 0 centered at 0 in H 3. Therefore for t > T 0 (ɛ) := log(r ɛ ), k(e s j) a t Ĉ 0 implies that k(e s j) K ɛ {(0, 1]j}, in other words, ka s K K ɛ A K. By the uniqueness of the left K-component, modulo the right multiplication by M, in the decomposition G = KA K, it follows that k K ɛ M, proving the second claim. For t R and T > 1, set K T (t) := {k K : a t k HKA + log T }. As a consequence of Proposition 4.4, we have the following. Corollary 4.5. (1) For all 0 t < log T, e K T (t). (2) For all t > log T, K T (t) =. (3) For any ɛ > 0, there exists T 0 (ɛ) 1 such that we have Thus for any T > 1, K T (t) K ɛ M for all t > T 0 (ɛ). (4.6) HKA + log T = 0 t<log T Ha t K T (t). Since B T (E) = H\HKA + log T N E, (4.6) together with Corollary 4.5 shows that B T (E) is essentially of the form H\Ha log T K ɛ MN E. The following proposition shows that B T (E) can be basically controlled by the set H\Ha log T N E. Proposition 4.7. Fix a bounded subset E of C. There exists l = l(e) 1 such that for all sufficiently small ɛ > 0, a t kmn z H lɛ ma t n z U lɛ holds for any m M, t > 0, z E, and k K ɛ. Proof. Recalling that N denotes the lower triangular subgroup of G, we note that the product map N A M N G is a diffeomorphism at

17 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 17 a neighborhood of e, in particular, bi-lipschitz. Hence there exists l 1 > 1 such that for all small ɛ > 0, (4.8) K ɛ N l 1 ɛ A l 1 ɛm l1 ɛn l1 ɛ. Similarly due to the H A N product decomposition of G ɛ, there exists l 2 > 1 such that (4.9) U ɛ H l2 ɛa l2 ɛn l2 ɛ for all small ɛ > 0 ([8, Lem 2.4]). We also have l 3 > 1 such that for all small ɛ > 0, (4.10) A (l1 +l 2 )ɛn (l1 +l 2 )ɛm l1 ɛ U l3 ɛ. Now let t > 0, k K ɛ, m M, n N. Then by (4.8), we may write k = n 1 b 1m 1 n 1 N l 1 ɛ A l 1 ɛm l1 ɛn l1 ɛ. Since a t n 1 a t N ɛ for t > 0, we have, by (4.9), Therefore a t n 1 a t = h 2 b 2 m 2 n 2 H l2 ɛa l2 ɛm l2 ɛn l2 ɛ. a t kmn = (a t n 1 a t)(a t b 1 m 1 n 1 )mn = (h 2 b 2 m 2 n 2 )a t b 1 m 1 n 1 mn = h 2 b 2 m 2 (a t b 1 b 1 1 a t)n 2 a t b 1 m 1 n 1 mn = h 2 a t (b 2 m 2 )b 1 (b 1 1 a tn 2 a t b 1 )m 1 n 1 mn h 2 a t A (l1 +l 2 )ɛm l2 ɛn (l1 +l 2 )ɛm l1 ɛmn by (4.10) h 2 a t U l3 ɛmn. As E is bounded, there exists l = l(e) > l 2 such that for all small ɛ > 0 and for all z E, U l3 ɛmn z mn z U lɛ. Since a t commutes with m, we obatin for all z E that a t kmn z H lɛ ma t n z U lɛ. Proof of Theorem 4.3. Let l = l(e) 1 be as in Proposition 4.7. For ψ C c (\G) and ɛ > 0, we define ψ ɛ ± C c (\G), ψ + ɛ (g) := sup u U lɛ ψ(gu) and ψ ɛ (g) := inf u U lɛ ψ(gu). For a given η > 0, there exists ɛ = ɛ(η) > 0 such that for all g \G, by the uniform continuity of ψ. ψ + ɛ (g) ψ ɛ (g) η

18 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 18 On the other hand, by Theorem 2.5, we have T 1 (η) 1 such that for all t > T 1 (η), (4.11) ψ ɛ + (ha t n)dh h H\H = s H\C 0 m M = (1 + O(η)) sk (C 0 ) m BMS mbr ψ ɛ + (sa t mn)dmdµ Leb (s) C 0 (ψ + ɛ,n)e (δ 2)t where ψ + ɛ,n(g) = m M ψ+ ɛ (gmn)dm. As N E is relatively compact, the implied constant can be taken uniformly over all n N E. Let T 0 (ɛ) > T 1 (η) be as in Proposition 4.4. For [e] = H H\G and s > 0, set so that Setting V T (s) := s t<log T [e]a t K T (t)n E B T (E) = V T (s) (B T (E) V T (s)). ψ H (g) := h H\H ψ(hg)dh, note that ψ H is left H-invariant as dh is a Haar measure. We will show that 1 lim sup T T δ ψ H sk (C 0 ) (g)dλ(g) = (1+O(η)) δ m BMS (ψ n )dn. [g] V T (T 0 (ɛ)) By Corollary 4.5, we have V T (T 0 (ɛ)) T0 (ɛ) t log T [e]a t K ɛ MN E. m BR n N E Let [g] V T (T 0 (ɛ)), so [g] = [e]a t kmn with T 0 (ɛ) t log T, k K ɛ, m M and n N E. By Proposition 4.7, there exist h 0 H and u U lɛ such that a t kmn = h 0 ma t nu so that [g] = [e]a t nu, since M H. We have ψ H (g) = ψ(ha t nu)dh h H\H h H\H ψ + ɛ (ha t n)dh. The measure e 2t dtdn is a right invariant measure of AN and [e]an is an open subset in H\G. Hence dλ(a t n) (restricted to [e]an) and e 2t dtdn are constant multiples of each other. It follows from the formula of dg that dλ(a t n) = e 2t dtdn. Therefore ψ H (g)dλ(g) ψ ɛ + (ha t n)dhe 2t dtdn. [g] V T (T 0 (ɛ)) n N E T 0 (ɛ)<t log T h H\H

19 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 19 By the choice of ɛ = ɛ(η), we also have m BR (ψ ɛ,n) + = (1 + O(η))m BR (ψ n ) where the implied constant depends only on ψ. Hence by (4.11), ψ ɛ,n(ha + t )dhe 2t dtdn n N E Hence lim sup T T 0 (ɛ)<t<log T h H\H sk (C 0 ) = (1 + O(η)) δ m BMS m BR (ψ n )dn (T δ e δ T 0 (ɛ) ). n N E 1 T δ [g] V T (T 0 (ɛ)) ψ H sk (C 0 ) (g)dλ(g) = (1+O(η)) δ m BMS m BR n N E (ψ n )dn. On the other hand, since \H is a closed subset of \G, so is 0 t s \Ha t KN E for any fixed s > 0; in particular, its intersection with a compact subset of \G is compact. Since [g] BT (E) V T (s)\hg 0 t s \Ha t KN E, and ψ has compact support, we have, as T, ψ(hg)dhdλ(g) = O(1). Therefore lim sup T 1 T δ [g] B T (E) V T (T 0 (ɛ)) [g] B T (E) h H\H ψ H sk (C 0 ) (g)dλ(g) (1+O(η)) δ m BMS As η > 0 is arbitrary and ɛ(η) 0 as η 0, we have 1 lim sup T T δ ψ H (g)dλ(g) sk (C 0 ) δ m BMS [g] B T (E) Similarly we can show that 1 lim inf T T δ ψ H (g)dλ(g) sk (C 0 ) δ m BMS [g] B T (E) 5. On the measure ω m BR n N E m BR n N E m BR n N E (ψ n )dn. (ψ n )dn. (ψ n )dn. In this section we will describe a measure ω on C and show that the term m BR n N E (Ψ n ) dn, which appears in the asymptotic expression in Theorem 4.3, converges to ω (E) as the support of Ψ shrinks to [e] with \G Ψ dg = 1. We keep the notations G, K, M, A +, N, N, a t, n z, n z, etc., from section 2. Throughout this section, we assume that is a non-elementary discrete

20 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 20 subgroup of G. Recall that {ν x = ν,x : x H 3 } denotes a -invariant conformal density for of dimension δ > 0. Definition 5.1. Define a Borel measure ω on C as follows: for ψ C c (C), ω (ψ) = e δ β z(x,z+j) ψ(z)dν,x (z) z C for x H 3 and z + j := (z, 1) H 3. In order to see that the definition of ω is independent of the choice of x H 3, we observe that for any x 1, x 2 H 3 and z C, e δ (β z(x 1,z+j) β z(x 2,z+j)) dν x 1 dν x2 (z) = e δ β z(x 1,x 2 ) dν x 1 dν x2 (z) = 1 by the conformality of {ν x }. Lemma 5.2. For any x = p + rj H 3 and ψ C c (C), ω (ψ) := (r 1 z p 2 + r) δ ψ(z)dν x (z). z C Proof. It suffices to show that β z (p + rj, z + j) = log z p 2 + r 2. r We use the fact that the hyperbolic distance d on the upper half space model of H 3 satisfies for z i + r i j H 3 (cf. [6]). Note that Now and hence cosh(d(z 1 + r 1 j, z 2 + r 2 j)) = z 1 z r r2 2 2r 1 r 2 β z (z, z + j) = β 0 (j, z + p + rj) = lim t t d( z + p + rj, e t j) = lim t t d(p + rj, z + e t j). cosh d(p + rj, z + e t j) = et ( z p 2 + r 2 ) + e t e d(p+rj,z+e tj) + e d(p+rj,z+e tj) = et ( z p 2 + r 2 ) + e t. r Therefore as t, Hence 2r d(p + rj, z + e t j) t + log z p 2 + r 2. r β z (p + rj, z + j) = log z p 2 + r 2. r

21 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 21 Definition 5.3. For a function ψ on C with compact support, define a function R ψ on MAN N G by R ψ (ma t n x n z ) = e δ t ψ( z) for m M, t R, x, z C. If ψ is the characteristic function of E C, we put R E = R χe. Since the product map M A N N G has a diffeomorphic image, the above function is well-defined. Proposition 5.4. For any ψ C c (C), ω (ψ) = k K/M R ψ (k 1 )dν j (k(0)). Proof. If k K with k 1 = ma t n x n z MAN N, since MAN fixes 0, k(0) = n z (0) = z. We note that lim s a s (j) = 0 and compute and hence 0 = β z (k(j), j) = β z (n z n x a tj, j) = β 0 (n x a tj, n z (j)) = lim s d(n x a tj, a s j) d(n z (j), a s j) = lim s d((a s n x a s)a s t j, j) d(n z (j), a s j) = lim s d(a s t j, j) d(n z (j), a s j) = lim s s t d(n z (j), a s j) t = lim s d(n z (j), a s j) s = β 0 (n z (j), j) = β z (j, z + j). Hence for k 1 K MAN N, R ψ (k 1 ) = e δ β k(0) (j,n k(0) (j)) ψ(k(0)).

22 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 22 Since the complement of NN AM/M in K/M is a single point and ν j is atom-free, we have k K/M = = = = R E (k 1 )dν j (k(0)) k (K NN AM)/M z C z C z C R E (k 1 )dν j (k(0)) e δ β k(0) (j,k(0)+j) ψ(k(0))dν j (k(0)) e δ β z (j, z+j) ψ( z)dν j ( z) e δ β z(j,z+j) ψ(z)dν j (z) = ω (ψ). Lemma 5.5. If (ma t n x n z )(m 1 a t1 n x 1 n z1 ) = m 0 a t0 n x 0 n z0 coordinates, then t 0 = t + t log( 1 + e t 1 x 1 z ) for some z C with z = z. in the MAN N Proof. Note that if m 1 = diag(e iθ 1, e iθ 1 ), then a t n x n z m 1 = m 1 a t n e iθ 1 x n e iθ 1 z. Hence we may assume m 1 = m = e without loss of generality. We use the following simple identity for z, x C: ( ) (5.6) n z n 1 + xz 0 x = 0 (1 + xz) 1 n x(1+xz) n z(1+xz) 1. Hence we have (a t n x n z )(a t1 n x 1 n z1 ) = (a t+t1 )(a 1 t 1 = a t+t1 n e t 1 x n e t 1 zn x 1 n z1 = a t+t1 n e t 1 x n x a t1 )(a 1 t 1 n z a t1 )n x 1 n z1 ( 1 + e t 1 x 1 z 0 0 (1 + e t 1 x 1 z) 1 ) n x 1 (1+e t 1 x 1 z) n e t 1 z(1+e t 1 zx 1 ) 1n z 1 = ma t+t1 +2 log( 1+e t 1 x 1 z )n x 2 n z2 for appropriate m M and x 2, z 2 C. Let E C be a bounded subset and U ɛ G a symmetric ɛ-neighborhood of e in G. For ɛ > 0, set (5.7) E + ɛ := U ɛ (E) and E ɛ := u Uɛ u(e).

23 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 23 Lemma 5.8. There exists l > 0 such that for all small ɛ > 0 and any g U lɛ, R E (k 1 g)dν j (k(0)) = (1 + O(ɛ)) ω (E ɛ ± ) k K/M where the implied constant depends only on E. Proof. Write k 1 = ma t n x n z and g = m 1 a t1 n x 1 n z1 U ɛ. By Lemma 5.5, we have k 1 g = m 0 a t0 n x0 n z0 where t 0 = t + t log( 1 + e t 1 x 1 z ). Since R E (k 1 g) = e δ t 0 χ E (g 1 k(0)), we have R E (k 1 g)dν j (k(0)) k K/M = = k(0) g(e) k(0) g(e) = (1 + O(ɛ)) e δ t 0 dν j (k(0)) e δ t e δ(t 1+2 log( 1+e t 1 x 1 z )) dν j (k(0)) k(0) E ± ɛ e δ t e δ(t 1+2 log( 1+e t 1 x 1 z )) dν j (k(0)). Since t 1 = O(ɛ), x 1 = O(ɛ) and z = k(0) g(e) E + ɛ, t log( 1 + e t 1 x 1 z ) = O(ɛ) where the implied constant depends only on E. Hence R E (k 1 g)dν j (k(0)) k K/M = (1 + O(ɛ)) = (1 + O(ɛ)) k(0) E ± ɛ k K = (1 + O(ɛ)) ω (E ± ɛ ). e δ t dν j (k(0)) R E ± ɛ (k 1 )dν j (k(0)) For ɛ > 0, let ψ ɛ be a non-negative continuous function in C(G) with support in U ɛ with integral one and Ψ ɛ C c (\G) be the -average of ψ ɛ : Ψ ɛ (g) := γ ψ ɛ (γg). We define Ψ ɛ E C c(\g) M by Ψ ɛ E(g) := z E m M Ψ ɛ (gmn z )dmdz. Lemma 5.9. For a bounded Borel subset E C, there exists c = c(e) > 1 such that for all small ɛ > 0, (1 c ɛ) ω (E ɛ ) m BR (Ψ ɛ E) (1 + c ɛ) ω (E + ɛ ).

24 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 24 Proof. Note that N is the expanding horospherical subgroup for the right action of a t, i.e., N = {g G : a t ga t e as t }. We have for ψ C c (G) M, m BR (ψ) = ψ(ka t n )e δt dndtdν j (k(0)) KAN (cf. [19, 6.2]). We note that d(a t n x mn z ) = dtdxdmdz is the restriction of the Haar measure dg to AN N G/M. We deduce m BR (Ψ ɛ E) = m BR (ψn ɛ z )dz z E = ψ ɛ (ka t n x mn z )e δt dmdxdtdν j (k(0))dz z E KAN m M = ψ ɛ (k(a t n x mn z ))χ E (z)e δt dxdtdmdzdν j (k(0)) k K AN MN = ψ ɛ (kg)r E (g)dgdν j (k(0)) k K g G ( = ψ ɛ (g) g U ɛ k K ) R E (k 1 g)dν j (k(0)) dg. Hence by Lemma 5.8 and the identity U ɛ ψ ɛ dg = 1, we have m BR (Ψ ɛ E) = (1 + O(ɛ))ω (E ± ɛ ). Corollary If ω ( (E)) = 0, then ω (E) = lim m BR ɛ 0 (Ψ ɛ E). Proof. For any η > 0, there exists ɛ = ɛ(η) such that ω (E + ɛ E ɛ ) < η. Together with Lemma 5.9, it implies that m BR (Ψ ɛ E) = (1 + O(ɛ))(1 + O(η))ω (E) and hence the claim follows. 6. Conclusion: Counting circles Let < G := PSL 2 (C) be a non-elementary discrete group with m BMS <. Suppose that P := (C) is a locally finite circle packing. Recall that sk (P) = sk (C) := e δ β (x,s) s+ dν,x (s + ), s Stab (C )\C where C is the set of unit normal vectors to Ĉ. It follows from the conformal property of {ν,x } that sk (C) is independent of the choice of C (C), and hence is an invariant of the packing (C). Theorem 1.6 is an immediate consequence of the following statement.

25 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 25 Theorem 6.1. Suppose that sk (C) <. For any bounded Borel subset E of C with ω ( (E)) = 0, we have N T (P, E) (6.2) lim T T δ Moreover sk (C) > 0 if P is infinite. = sk (P) δ m BMS ω (E). The second claim on the positivity of sk (C) follows from the second claim of Theorem 2.5 and Lemma 3.1. We will first prove Theorem 6.1 for the case when C is the unit circle C 0 centered at the origin and deduce the general case from that. The case of C = C 0. Fix η > 0. As ω ( (E)) = 0, there exists ɛ = ɛ(η) > 0 such that (6.3) ω (E + 4ɛ E 4ɛ ) η where E 4ɛ ± is defined as in (5.7): E+ 4ɛ := U 4ɛ(E) and E4ɛ := u U 4ɛ u(e). We can find a P-admissible Borel subset Ẽ+ ɛ such that E Ẽ+ ɛ E ɛ + by adding all the open disks inside E ɛ + intersecting the boundary of E. Similarly we can find a P-admissible Borel subset Ẽ ɛ such that Eɛ Ẽ ɛ E by adding all the open disks inside E intersecting the boundary of Eɛ. By the local finiteness of P, there are only finitely many circles intersecting E (resp. Ẽɛ ) which are not contained in Ẽ+ ɛ (resp. E). Therefore there exists q ɛ 1 (independent of T ) such that (6.4) N T (P, Ẽ ɛ ) q ɛ N T (P, E) N T (P, Ẽ+ ɛ ) + q ɛ. Recalling the set B T (Ẽ± ɛ ) = H\HKA + log T N Ẽ± ɛ Proposition 3.7 and (6.4) that for all T 1, H\G, it follows from (6.5) #[e] B T (Ẽ ɛ ) m 0 N T ((C 0 ), E) #[e] B T (Ẽ+ ɛ ) + m 0 for some fixed m 0 = m 0 (ɛ) 1. Lemma 6.6. There exists l > 0 such that for all T > 1 and for all small ɛ > 0, KA + log T U ɛ KA + log T +ɛ N lɛ where N lɛ is the lɛ-neighborhood of e in N. Proof. We may write U ɛ = M ɛ N ɛ A ɛ N ɛ = K ɛ A ɛ N ɛ up to uniform Lipschitz constants. For u = mn an M ɛ N ɛ A ɛ N ɛ, a t u = m(a t n a t )a t an. Since a t n a t U ɛ for t > 0, we may write it as k 1 a 1 n 1 K ɛ A ɛ N ɛ. Hence for 0 < t < log T, we have (a 1 a t n 1 a t a) N ɛ and a t u = (mk 1 )(a 1 a t a)(a 1 a t n 1 a t a)n KA + log T +2ɛ N 2ɛ. This proves the claim.

26 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 26 Lemma 6.7 (Stability of KAN-decomposition). There exists l 0 > 0 (depending on E) such that for all T > 1 and for all small ɛ > 0, KA + log T N Ẽ+ ɛ U l 0 ɛ KA + log T +ɛ N E + 2ɛ; KA + log T ɛ N E 2ɛ KA + log T ( u U l0 ɛ N Ẽ ɛ u). Proof. There exists l 0 > 0 depending on E such that N Ẽ ɛ + U l 0 ɛ U ɛ N E +. 2ɛ Hence the first claim follows from Lemma 6.6. The second claim can be proved similarly. For ɛ > 0, define functions F ɛ,± T F ɛ,+ T (g) := χ Be ɛ T (N E + )([e]γg); 2ɛ γ (H )\ on \G: F ɛ, T (g) := γ (H )\ χ Be ɛ T (N E 2ɛ )([e]γg). Let l 0 be as in Lemma 6.7. Without loss of generality, we may assume that l 0 < l for l as in Lemma 5.8. Lemma 6.8. For all g U l0 ɛ and T 1, (6.9) F ɛ,+ T (g) m 0 N T ((C 0 ), E) F ɛ,+ T (g) + m 0. Proof. Note that, since U l0 ɛ is symmetric, for any g U l0 ɛ, #[e] B T (Ẽ+ ɛ ) #[e] B T (Ẽ+ ɛ )U l0 ɛg 1 #[e]g B e ɛ T (N E + ), 2ɛ by Lemma 6.7, which proves the second inequality by (6.5). The other inequality can be proved similarly. For ɛ > 0, let ψ ɛ be a non-negative continuous function in C(G) with support in U l0 ɛ with integral one and Ψ ɛ C c (\G) be the -average of ψ ɛ : Ψ ɛ (g) := γ ψ ɛ (γg). By integrating (6.9) against Ψ ɛ, we have Since F ɛ, T, Ψɛ m 0 N T ((C 0 ), E) F ɛ,+ T, Ψɛ + m 0. F ɛ,+ T, Ψɛ = = = \G γ H\ g H\G [g] B e ɛ T (N E + 2ɛ ) χ Be ɛ T (N E + 2ɛ )([e]γg)ψ ɛ (g) dg χ Be ɛ T (N E + 2ɛ )([e]g)ψ ɛ (g) dg h H\H Ψ ɛ (hg) dhdλ(g) we deduce from Theorem 4.3 and Lemma 3.2 that (6.10) F ɛ,+ T, Ψɛ sk (C 0 ) δ m BMS m BR (Ψ ɛ n)dn T δ e ɛδ n N E + 2ɛ

27 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 27 where Ψ ɛ n(g) = m M Ψɛ (gmn)dm. Therefore by applying Lemma 5.9 to (6.10) and using (6.3), we deduce F ɛ,+ T lim sup, Ψɛ sk (C 0 ) T T δ (1 + ɛ) (Ψ ɛ n)dn δ m BMS m BR n N E + 2ɛ sk (C 0 ) (1 + ɛ)(1 + cɛ) δ m BMS ω (E 4ɛ + ) (1 + c 1 η)(1 + c 2 ɛ) sk ((C 0 )) δ m BMS ω (E) where the constants c, c 1, c 2 depend only on E. Similarly, we have lim inf T F ɛ,+ T, Ψɛ T δ sk (C 0 ) (1 c 1 η)(1 c 2 ɛ) δ m BMS ω (E). As η > 0 is arbitrary and ɛ = ɛ(η) 0 as η 0, we have N T ((C 0 ), E) lim T T δ This proves Theorem 6.1 for C = C 0. = sk (C 0 ) δ m BMS ω (E). The general case. Let r > 0 be the radius of C and p C the center of C. Set ( ) ( ) 1 p r g 0 = n p a log r = r 1. Then g0 1 (z) = r 1 (z p) for z C and g0 1 (C) = C 0. Setting 0 = g0 1 g 0, we have N T ((C), E) = #{C (g 0 (C 0 )) : C E, Curv(C) < T } = #{g 1 0 (C) 0(C 0 ) : C E, Curv(C) < T } = #{C 0 (C 0 ) : C g 1 0 (E), Curv(C ) < r 1 T } = N r 1 T ( 0 (C 0 ), g 1 0 (E)). We claim that (6.11) 1 m BMS 0 sk 0 ( 0 (C 0 )) r δ ω 0 (g0 1 (E)) = 1 m BMS sk ((C)) ω (E). Note that the each side of the above is independent of the choices of conformal densities of 0 and respectively. Fixing a -invariant conformal density {ν,x } of dimension δ, set ν 0,x := g 0 ν,g0 (x)

28 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 28 where g 0 ν,g 0 (x)(r) = ν,g0 (x)(g 0 (R)). It is easy to check that ν 0,x is supported on Λ( 0 ) = g 0 Λ() and satisfies dν 0,x (z) = e δ β z(x,y) ; dν 0,y γ ν 0,x = ν 0,γ(x) for all x, y H 3, γ 0 and z Ĉ. Hence {ν 0,x : x H 3 } is a 0 -invariant conformal density of dimension δ = δ 0 and satisfies that for f C c (C) f(z)dν 0,x(z) = f(g0 1 (z))dν,g 0 (x)(z). g 0 (z) E We consider the Bowen-Margulis-Sullivan measures m BMS and m BMS 0 on \ T 1 (H 3 ) and 0 \ T 1 (H 3 ) associated to {ν,x } and {ν 0,x}, respectively. Lemma For a bounded Borel function ψ on \ T 1 (H 3 ), consider a function ψ g0 on 0 \ T 1 (H 3 ) given by ψ g0 (u) := ψ(g 0 (u)). Then m BMS 0 In particular, m BMS 0 = m BMS. Proof. Note that if v = g(u), then Since ν 0,x = g 0 ν,g 0 (x), we have m BMS 0 (ψ g0 ) = = u 0 \ T 1 (H n ) v \ T 1 (H n ) = m BMS (ψ). Similarly, we can verify: z E (ψ g0 ) = m BMS (ψ). β u ±(x, π(u)) = β v ±(g(x), π(v)). ψ(g 0 (u))e δ β u + (x,π(u)) e δ β u (x,π(u)) dν 0,x(u + )dν 0,x(u )dt ψ(v)e δ β v + (g 0 (x),π(v)) e δ β v (g 0 (x),π(v)) dν,g0 (x)(v + )dν,g0 (x)(v )dt Lemma For any x H 3, e δ β s + (x,s) dν 0,x(s + ) = e δ β s+ (g 0 (x),s) dν,g0 (x)(s + ); s Stab (C )\C s Stab 0 (C 0 )\C 0 that is, sk ((C)) = sk 0 ( 0 (C 0 )). Lemma For any bounded Borel subset E C, ω 0 (g 1 0 (E)) = rδ ω (E).

29 ASYMPTOTIC DISTRIBUTION OF CIRCLES IN ORBITS OF KLEINIAN GROUPS 29 Proof. Since g0 1 (z) = r 1 (z p), r is the linear distortion of the map g0 1 in g the Euclidean metric, that is, r = lim 1 0 (w) g 1 0 (w 0) w w0 w w 0 for any w 0 C. Hence ( w 2 + 1) δ dν,g0 (j)(w) = r δ ( g0 1 (w) 2 + 1) δ Since ν 0,x = g0 ν,g 0 (x), we deduce dν,j (w). ω 0 (g0 z g 1 (E)) = ( z 2 + 1) δ dν 0,j(z) 1 0 (E) = ( g0 1 (u) 2 + 1) δ dν,g0 (j)(u) u E = r δ ( u 2 + 1) δ dν,j (u) u E = r δ ω (E). This concludes a proof of (6.11). Therefore, since sk 0 (C 0 ) < and m BMS 0 <, the previous case of C = C 0 yields that 1 lim T T δ N 1 T ((C), E) = lim T T δ N r 1 T ( 0 (C 0 ), g0 1 (E)) 1 = δ 0 m BMS 0 sk 0 (C 0 ) r δ ω 0 (g0 1 (E)) 1 = δ m BMS sk (C) ω (E). This completes the proof of Theorem 6.1. References [1] Lars V. Ahlfors. Finitely generated Kleinian groups. Amer. J. Math., 86: , [2] Rufus Bowen. Periodic points and measures for Axiom A diffeomorphisms. Trans. Amer. Math. Soc., 154: , [3] David W. Boyd. The residual set dimension of the Apollonian packing. Mathematika, 20: , [4] Marc Burger. Horocycle flow on geometrically finite surfaces. Duke Math. J., 61(3): , [5] Schieffelin Claytor. Topological immersion of Peanian continua in a spherical surface. Ann. of Math. (2), 35(4): , [6] J. Elstrodt, F. Grunewald, and J. Mennicke. Groups acting on hyperbolic space. Springer Monographs in Mathematics. Springer-Verlag, Berlin, Harmonic analysis and number theory. [7] L. Flaminio and R. J. Spatzier. Geometrically finite groups, Patterson-Sullivan measures and Ratner s rigidity theorem. Invent. Math., 99(3): , [8] Alex Gorodnik, Nimish Shah, and Hee Oh. Strong wavefront lemma and counting lattice points in sectors. Israel J. Math, 176: , 2010.